Patrons of a local bookstore can sign up for advance notification of new book arrivals in genres of interest. In the first month of this service, 32 sign up for mysteries, 34 for spy novels, 18 for westerns, and 41 for science fiction. Of these, 17 sign for both mysteries and spy novels, 8 for both mysteries and westerns, 19 for mysteries and science fiction, 5 for spy novels and westerns, 20 for spy novels and science fiction, and 12 for westerns and science fiction. In addition, 2 sign up for mysteries, spy novels and westerns; 11 for mysteries, spy novels, and science fiction; 6 for mysteries, westerns, and science fiction; and 5 for spy novels, westerns, and science fiction. Finally, 2 people sign up for all four categories. How many people signed up for service in the first month?

for all n greater than or equal to 0. You can prove this without induction, just by simplifying the above formula and using the fact that 0! = 1. But, to prove this by induction, look at the case of n = 0, then assume that it's true for n = k, and prove for n = k + 1.

n = 0:

The statement becomes:

0!/(0 - 0)!(0!) = 0!/(0 - 0)!(0!) = 1
1/1*1 = 1/1*1 = 1
1 = 1 = 1

Assuming the statement is true for n = k, prove it is true for n = k + 1:

Using k = k -0 and 0 = k - k, the previous statement can be written as k!/(k - 0)!(0!) = k!/(k - k)!(k!) = 1, which is the statement when n = k. Because we're assuming that the statement is true for n = k, we have also shown that it is true for n = k + 1.